Monkey Saddle (n-fold)
Monkey Saddle (n-fold) is an algebraic surface, defined by an implicit equation, embedded.
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algebraic aperiodic def-implicit embedded implemented noncompact tradition-classical
About
An ordinary saddle rises in two directions and falls in two. This one falls in three -- two for the legs and one for the tail, which is where the name comes from. It is the standard picture of a degenerate critical point: the surface is flat to second order at the origin, so the usual test for a maximum or minimum tells you nothing there.
Formula
Properties
- Family
- algebraic
- Given by
- an implicit equation
- Curvature
- no condition imposed
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- embedded
- Orientable
- yes
- Fidelity
- exact
- Exactness
- elementary
Sources
Closed form reproduced numerically against the shipped implementation (math_art/surfaces/algebraic.py) over 240 sample points: matches oracle x -1 over 240 points (worst 4.44e-16).
- The standard example of a degenerate critical point of higher order; z = x^3 - 3xy^2 is the real part of z^3.